(1) Field of the Invention
The present invention relates to computer model of hydrodynamic flows and more particularly, relates to modeling partially cavitating flows over a supercavitating axisymmetric body.
(2) Description of the Prior Art
Modeling of boundary flows about objects subject to laminar and turbulent flows is well known in the art. High speed underwater vehicles, however, cause cavitation of the surrounding fluid. Cavitation reduces pressure in the fluid below its vapor pressure causing the fluid to vaporize, allowing the undersea vehicle to travel with lower friction when the vehicle is completely surrounded by the cavity.
Partial cavitation is an unsteady phenomenon that occurs when part of the supercavitating vehicle is traveling in the cavity. Specifically, this phenomenon occurs during launch of the vehicle. A steady, partial cavitation allows development of vehicle designs which take advantage of drag reduction through cavitation. It may also be possible to take advantage of drag reduction with partial cavitation by properly directing the re-entrant jet that forms in the cavity closure region. Partial cavitation often occurs during maneuvering of the supercavitating vehicle.
A slender body theory has been developed to solve axisymmetric supercavitating flows. Using the slender body method, sources are defined along the body-cavity axis and control points along the body-cavity surface. A nonlinear differential equation is formed by imposing dynamic boundary conditions on the cavity. A conical cavity closure is assumed in order to solve the developed nonlinear differential equation.
A non-linear boundary element method for determining a cavity shape has been developed. Source and dipole strengths along the body-cavity surface are determined using kinematic boundary conditions on the wetted body surface and dynamic boundary conditions on the assumed cavity shape. The kinematic boundary condition is then used to update the cavity shape. The process is then iterated to solve for the unknown cavity shape.
Two numerical hydrodynamics models have been developed by the Naval Undersea Warfare Center for axisymmetric super cavitating high speed bodies. These models are the slender body theory (SBT) model and the boundary element (BE) model. Both of these models have been proven to predict cavity shape and parameters with good accuracy.
These models, however, do not account for the transition case when the vehicle is subjected to only partial cavitation.
In the SBT model, total drag is predicted by adding the pressure drag obtained from the model solution and the viscous drag obtained by applying the Thwaites and Falkner-Skan approximations along the wetted portions of the cavitator. This method is extended to subsonic compressible flows using the compressible Green's function. In the BE model, sources and dipoles are defined on the body-cavity shape and are solved using Green's formula. This yields a Fredholm integral equation of the second kind which gives the supercavitating cavity shape.
Partial cavitation modeling has been done by Uhlman, J. S. (1987), The Surface Singularity Method Applied to Partially Cavitating Hydrofoils, Journal of Ship Research, Vol. 31, No. 2, pp. 107-24; Uhlman, J. S. (1989), The Surface Singularity or Boundary Integral Method Applied to Supercavitating Hydrofoils, Journal of Ship Research, Vol. 33, No. 1, pp. 16-20; Kinnas, S. A., and Fine, N. E. (1990), Non-Linear Analysis of the Flow Around Partially and Super-Cavitating Hydrofoils by a Potential Based Panel Method, Proceedings of the IABEM-90 Symposium, International Association for Boundary Element Methods, Rome, Italy, and Kinnas, S. A., and Fine, N. E. (1993), A Numerical Nonlinear Analysis of the Flow Around Two- and Three-Dimensional Partially Cavitating Hydrofoils, Journal of Fluid Mechanics, Vol. 254. However, these methods are explicitly adapted for hydrofoils, and the theories presented therein are not readily adapted to supercavitating vehicles.